Generalized skew-derivations acting on multilinear polynomial in prime rings

Abstract

Let $\mathcal{R}$ be a noncommutative prime ring of characteristic different from 2, ${\mathcal{Q}}_r$ be the right Martindale quotient ring of $\mathcal{R}$ and $\mathcal{C}=Z({\mathcal{Q}}_r)$ be the extended centroid of $\mathcal{R}$. Suppose that $\pi (x_1,\dots ,x_n)$ is a noncentral multilinear polynomial over $\mathcal{C},\mathcal{S}=\{\pi (r_1,\dots ,r_n)|r_1,\dots ,r_n\in \mathcal{R}\},\mathcal{F},\mathcal{G}$ and $\mathcal{H}$ are three generalized skew-derivations of $\mathcal{R}$ associated to the same automorphism α. Let $h,f,g$ be the associated skew derivations respectively of $\mathcal{H},\mathcal{F}$ and $\mathcal{G},$ such that h, f, g are commuting with α. We will provide the complete description of $\mathcal{H},\mathcal{F}$ and $\mathcal{G}$, in the case $\mathcal{F}(X)\mathcal{G}(X)=\mathcal{H}(X^2),$ for all $X\in \mathcal{S}$.

Keywords:

• Extended centroid
• generalized skew derivations
• Martindale right ring of quotient
• multilinear polynomials
• prime rings

2020 MATHEMATICS SUBJECT CLASSIFICATION::

• 16N60
• 16W25

Acknowledgments

The authors would like to express their sincere thanks to the reviewers and referees for the constructive comments and suggestions which help to improve the quality of the paper.

Additional information

Funding

The first author is supported by DST-INSPIRE (IF190599), Ref. No. DST/INSPIRE/03/2019/000711.